What I don't understand is how a pair of sets ( which are infinite ) constitutes a real number.

It is difficult to understand, but your interpretation is correct. The author intends to define "a real number" as a pair of sets of rational numbers that have certain properties. The "game" that is being played is to define real numbers in terms of structures involving the rational numbers.

Intuitively, you can think of this approach as establishing a real number like ##\sqrt{2}## as boundary between two sets of rational numbers ( those less than ##\sqrt{2}## and those greater), but you won't be able to follow the formal proofs unless you take the definition given for a real number literally - as a structure involving two sets of rational numbers.

The development of the real numbers via Dedekind cuts is a chapter in most books on real analysis, but it is a chapter that isn't used in later chapters. People negotiate the later chapters by thinking about real numbers the way they thought about them before they read about Dedekind cuts.

Then few lines below he make the following definition:
A real number is a cut in ##\mathbb{Q}##.

What I don't understand is how a pair of sets ( which are infinite ) constitutes a real number.
I would like somebody to give me, if possible, a brief explanation about Dedekind cuts, too.

Don't know if this helps but remember "infinity" is no specific number or formula..merely an idea or concept to help us deal with certain metaphysical/maths ideas..I.e. "the infinite" is just a very useful..idea.

"All the points in A are less than all the points of B. The real number is the boundary between A and B, which is in A or B, but not both."

This is not correct for two separate reasons. A and B constitute a partition of the rational numbers ℚ into two sets, but condition 3) requires that A contain no largest element.

So A never contains the real number that the Dedekind cut represents, even if it is rational; in that case the represented number lies in B. But if the represented real number is irrational (for example, if A is all rationals less than √2 and B is all rationals greater than √2), then neither A nor B contains the represented number.

"All the points in A are less than all the points of B. The real number is the boundary between A and B, which is in A or B, but not both."

This is not correct for two separate reasons. A and B constitute a partition of the rational numbers ℚ into two sets, but condition 3) requires that A contain no largest element.

So A never contains the real number that the Dedekind cut represents, even if it is rational; in that case the represented number lies in B. But if the represented real number is irrational (for example, if A is all rationals less than √2 and B is all rationals greater than √2), then neither A nor B contains the represented number.

You are right. I should have said "may be in A or B, but not both". If the cut is a rational number, it will be in A or B. If it is irrational, neither.

The real numbers are the cuts, which are pairs of sets. None of them appear in A or B for any cut.

The cut is a rational real number if it is the image of some ##q## in the set ##\mathbb{Q}## of rationals from which the reals are constructed under the mapping ##\phi(q)=\{A,B\}## and ##A=\{x\in\mathbb{Q}:x<q\}##, ##B=\{x\in\mathbb{Q}:x\geq q\}## (according to the definition).

Note that in this case ##q## (not the rational real image of ##q##) appears specifically in ##B##. We cannot accept both ##q\in A\wedge q\notin B## and ##q\in B\wedge q\notin A## as cuts, otherwise the system of reals constructed would have jumps at every rational real under its ordering ##C\leq C'## defined as ##A(C)\subseteq A(C')## (where ##A(X)## is the ##A## for cut ##X##).